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Sketch the region. V

Sketch the region. V

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Use the limit process to find the area of the region between the graph of the function and the y-axis over the given y-interval. \[ f(y) = 2y, \quad 0 \le y \le 2 \] Sketch the region. **Diagrams Explained:** 1. **Top-Left Graph:** - The function is linear with a positive slope, \( f(y) = 2y \). - The region shaded is a triangle with its base along the x-axis from \( f(0) = 0 \) to \( f(2) = 4 \) and height from \( y = 0 \) to \( y = 2 \). 2. **Top-Right Graph:** - The function appears to have a negative slope. - The shaded region is a right triangle, with its vertex on the y-axis at \( y = 0 \) extending leftward. 3. **Bottom-Left Graph:** - This graph is similar to the top-left with a triangle formed under the function. - The triangle is highlighted, showcasing the area calculation on the positive slope graph. 4. **Bottom-Right Graph:** - Similar to the top-right graph with a negative slope. - The shaded triangle is formed along the y-axis and extends horizontally onwards, likely part of a demonstration on symmetry or different interpretation. Each circle below the graphs suggests a selection, possibly for choosing which graph correctly represents the desired area.
Transcribed Image Text:Use the limit process to find the area of the region between the graph of the function and the y-axis over the given y-interval. \[ f(y) = 2y, \quad 0 \le y \le 2 \] Sketch the region. **Diagrams Explained:** 1. **Top-Left Graph:** - The function is linear with a positive slope, \( f(y) = 2y \). - The region shaded is a triangle with its base along the x-axis from \( f(0) = 0 \) to \( f(2) = 4 \) and height from \( y = 0 \) to \( y = 2 \). 2. **Top-Right Graph:** - The function appears to have a negative slope. - The shaded region is a right triangle, with its vertex on the y-axis at \( y = 0 \) extending leftward. 3. **Bottom-Left Graph:** - This graph is similar to the top-left with a triangle formed under the function. - The triangle is highlighted, showcasing the area calculation on the positive slope graph. 4. **Bottom-Right Graph:** - Similar to the top-right graph with a negative slope. - The shaded triangle is formed along the y-axis and extends horizontally onwards, likely part of a demonstration on symmetry or different interpretation. Each circle below the graphs suggests a selection, possibly for choosing which graph correctly represents the desired area.
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